Melt butter in a large pan. Add mushrooms, garlic, and a pinch of salt and sauté until mushrooms are softened and excess water has cooked off, 8-10 minutes. Remove from heat.
Combine ground flaxseed, Parmesan, chickpeas, dates, parsley, egg, fennel seeds, tahini, tamari, 1/2 teaspoon salt, and 1 teaspoon pepper in a food processor. Pulse to combine well and transfer to a large mixing bowl. Once mushrooms are at room temperature, add them, along with pan juices, to the bowl along with rice and stir to combine. Cover and refrigerate for about 30 minutes or up to 1 day. If the mixture seems too wet to form into a patty, stir in 1-2 tablespoons oats.
Preheat oven to 475° with rack in upper third.
Form the mixture into 6 patties, each 1" thick. Line a baking sheet with parchment paper and arrange the patties with space in between. Bake until toasted, 14-18 minutes.
Warm oil in a sauté pan over medium heat. Add shallots and a pinch of salt and sauté until edges begin to brown, 5-7 minutes. Set aside.
After removing the patties from the oven, toast the English muffins. Put a swipe of hummus on each muffin half and assemble by layering patty, avocado, arugula, and shallots.
